SAT-63: Margin of Error and Confidence Intervals

SAT-63: Margin of Error and Confidence Intervals

Description: A survey result like "62% ± 3%" carries a margin of error because a sample only estimates the true population value. This lesson explains how to read that range — the confidence interval — and what makes the margin larger or smaller. You interpret these on the SAT; you almost never compute them.

What the margin of error means

The margin of error tells you how far the true population value could reasonably be from your sample estimate. "62% ± 3%" means the true percentage is plausibly between 59% and 65%. That range is the confidence interval. (Oʻzbekcha: xatolik chegarasi — haqiqiy qiymat namuna natijasidan qancha uzoq boʻlishi mumkinligini koʻrsatadi.)

Reading a confidence interval

To build the interval, subtract and add the margin to the estimate: interval = estimate ± margin. Any value inside the interval is plausible for the true value; values outside it are unlikely. (Oʻzbekcha: ishonch oraligʻi = baho ± xatolik chegarasi.)

What changes the margin of error

• Bigger sample size → smaller margin. More data means a more precise estimate.
• Higher confidence level (e.g. 95% → 99%) → larger margin. To be more certain, you need a wider net.

So a small margin of error usually signals a large sample. (Oʻzbekcha: namuna qancha katta boʻlsa, xatolik chegarasi shuncha kichik boʻladi.)

Worked Example 1 — build the interval

A poll estimates 48% support with a margin of error of 4%. What is the confidence interval?

• 48% − 4% = 44%; 48% + 4% = 52%.
• Interval: 44% to 52% — the true support is plausibly in this range.

Worked Example 2 — interpret a claim

Using that 44%–52% interval, can the pollster be confident support is above 50%?

• No. Since values like 45% and 49% are inside the interval, the true value could be below 50%.
• The data does not support "a majority supports it."

(Oʻzbekcha: 50% dan past qiymatlar ham oraliqda boʻlgani uchun "koʻpchilik qoʻllaydi" deb boʻlmaydi.)

Worked Example 3 — compare two surveys

Survey A (500 people) reports ± 4%; Survey B (2,000 people) reports ± 2%. Which is more precise and why?

• Survey B has the smaller margin, so it is more precise — its larger sample narrows the interval.

Tip: to halve a margin of error you generally need a much larger sample, not just a slightly bigger one. (Oʻzbekcha: xatolikni kichraytirish uchun namunani sezilarli kattalashtirish kerak.)

What a confidence interval does NOT mean

Students often misread these. A "95% confidence interval of 44% to 52%" does not mean 95% of people chose something, and it does not mean there is a 95% chance the true value is exactly the estimate. It means the method is reliable enough that intervals built this way capture the true population value most of the time. For the SAT, the safe interpretation is simply: "the true value is plausibly anywhere in this range." Any answer choice that states a single exact value as certain is wrong. (Oʻzbekcha: ishonch oraligʻi — haqiqiy qiymat shu oraliqda boʻlishi ehtimoli yuqori degani; aniq bitta songa kafolat bermaydi.)

Two intervals that overlap

If two groups' confidence intervals overlap, you cannot conclude they are truly different — the difference might just be sampling error. If the intervals do not overlap, the difference is more convincing. This overlap idea is a common SAT reasoning question. (Oʻzbekcha: ikki oraliq kesishsa, farq ishonchli emas; kesishmasa, farq haqiqiyroq.)

Practice 1

An estimate is 30% with a margin of error of 5%. Give the confidence interval.

Practice 2

A researcher wants a smaller margin of error without lowering the confidence level. What should change?

Key words — Kalit soʻzlar

• Margin of error — xatolik chegarasi
• Confidence interval — ishonch oraligʻi
• Confidence level — ishonch darajasi
• Estimate — baho (taxminiy qiymat)
• Sample size — namuna hajmi
• Precise — aniq
• Plausible — ehtimoli bor
• True value — haqiqiy qiymat
• Population — populyatsiya

Summary

• Confidence interval = estimate ± margin of error; any value inside is plausible.
• Larger sample → smaller margin; higher confidence level → larger margin.
• You can only claim "above X" if the whole interval lies above X.